You already have an answer. As in the timetable by Hyperdia, Airport-Kaitoku does not pass the previous train to Haneda. The former 7:24 train arrives at Haneda at 7:39, and the latter 7:30 train gets to Haneda at 7:45. Airport-Kaito leaves on 7:28 and arrives at 7:41, just in between. There is a difference for which intervening stations the trains stop, but as far as a traveler to Haneda, you just have to confirm the train goes to Haneda. A few minutes difference does not matter. Just leave your hotel earlier.http://www.hyperdia.com/cgi/en/intervaltime.html?dep_node=HIGASHI-NIHO...http://www.haneda-tokyo-access.com/en/transport/trainlinemap.html
There is a legend for ト(To) and 急(Kyu) at the bottom of the page. To for Tokkyu(Limited Express) and Kyu for Kyuko(Express), but as I said, you do not have to care about it. This is a fairly frequent line. No train stops for a long time.
For a transfer from Bakuro-Yokoyama to Higashi-Nihombashi, the red line on the figure would help you.https://www.kotsu.metro.tokyo.jp/subway/stations/bakuro-yokoyama/i/sol...
Your train from Funabori bound for Shinjuku/新宿 comes from the bottom-right and stop at the green platform #1. You take an elevator to the top and get out of the gate. Your next train to Haneda comes from the top-left and arrives at the pink platform #1. There are three choices to get to that platform.
1) Use two elevators, A4 in the middle of the map and another near B1. This requires ground transportation, which includes a road crossing. You need to wait for a signal over a broad avenue.
2) Take an underpass, involving two stairs/escalators.
3) Take a train to another direction from platform #2. The next station is Asakusabashi. This small station has an island platform. The transfer to a train in the opposite direction is just easy.https://www.kotsu.metro.tokyo.jp/eng/services/subway/stations/asakusab...
As a traveler with the luggage, I would think of plan 3). To do so, I would leave earlier. The number of transfer would increase, but it is easier. If I could not leave early, I would take plan 2). If the weather is good enough, I might take plan 1) instead. This requires an earlier departure as in plan 3).